Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-4x+6y &= 8 \\ 6x-8y &= -7\end{align*}$
Explanation: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $3$ and the bottom equation by $2$ $\begin{align*}-12x+18y &= 24\\ 12x-16y &= -14\end{align*}$ Add the top and bottom equations. $2y = 10$ Divide both sides by $2$ and reduce as necessary. $y = 5$ Substitute $5$ for $y$ in the top equation. $-4x+6( 5) = 8$ $-4x+30 = 8$ $-4x = -22$ $x = \dfrac{11}{2}$ The solution is $\enspace x = \dfrac{11}{2}, \enspace y = 5$.